Question

A company produces x units of product A and y units of product B (both in hundreds per month). The monthly profit equation (in thousands of dollars) is given by the following: P(x,y) = -4x^2 + 4xy - 3y^2 + 4x + 10y +81 Find P: (x,y) and evaluate P,(1,3). Use formulas to get an exact answer! Px(x,y) = Px(1,3) = What does this value mean? Answer in this textbox. Answer: What is the critical point of P(x,y)? What are the equations for Prex (x,y), Pxy(x, y), and Pyy(x,y)? Critical point: Pxy(x,y) = Pxx(x,y) = Pyy(x,y) =

Answer 1

Here if you have any doubts please ask me in comments here i used genral langrage multipliers methods

* 10 Pla, y) = 4x2 + 4xy – 3y2 + 412 + loy + 81 Pac (2,4)= ap dx 8x + 4y +4. ļ Px11,3)= -8(1) + 4(3) + 4 8 Pa (x, y) = 8x + 4y + 4. Answer Px (1,3) 8 Answer Here P₂(x, y) means change of probit with respect to unit of product A wonen B constant inst Pa (1,3) means change of profit 3 unit of with respect to product B. Answer: of A product, P(x,y). (ii) for critical point of OP -O ax ду OP0, OP ox 8x + 4y + 4 = 0 OP dy +42-6y +10 - 8x + 4y + 4 = 0, → 8x -12y + 20 =0 x = 2 critical point (2, 3). y = 3 3 O O POW.9) 4 Now Pay(x,y) = dy dx Pax (x,y) = a dx ( P) -8 ; Pyy(x,y) = ( ) = -6, → Critical point .- 1x=2, y = 3, (2, 3) Pay(x,y) 4. Answer Paxlx,y) 8 Pyy (x,y) - 1-6 Now Pxa <-8, Pax Pyy - (Pxy)2 = 32 >o 2=2, Y-3 is maximum poine,

Pmaza -4(2)2 + 4(2)(3) : 3(3)2 + 4(2) +10(3) +81. -16 + 24 - 27 +8+ 30+81 100 2, y = 3, x = maximum profit = 100 $. Answer Answer Answe (0:2) (0.8) N(3,9) = 102 y x: Number of Labour : y Number of unit of capital rea. porodua; N. Unit (0-8)-1 y(0-2) x Nq (x,y) = ON дх 10(0.8) 0.2 - 8 8 40-2 2-0.2. 2()°8 2.6 0.2 2 5 1 Ny(x,y) NE dy 10 (0.2) 20-8 y0-8 0.2 - Nge ( 40, 50) = 8 () Ny (40,50) 2() 0.8 2.6 -0.2 = 2 5 22.65 0.2 Nx (40,50) = V Na (39) = 840.2 x-0.2 2 20-8 y-0-8 Ny 140,50) 22.05-0.2 Ny(2,4) Answer Labor given each unit n cost $ 100; each unit capital cost $50 e total 10,000 is budget 10000 100 x + 50 ya 2x+y = 200 g(x,y) = 100x+ soy -10000 Answer 0-8 yo.2 + hl 100x+ soy -10000). F(x,y,h) = 102 Answer 8x -0.2 40.2 + looh. > Fx (x, y, h) = Answer

Now af ah Fy (X14, h) = OF Fh (Hry, h) - ду Fy(X,4,5) = 2 xo8 g-0-8 + 50 b. Answer » Fn(x,y,hl 1002 + 5o y - 10000 0 Answer * * for maximization process,' Fx=0, Fy=0; 2x0.8 y-o. Вх -0.2 y0:2 +100h = 0 + 50 h = 0 4x0.8 y-0.8 8 x-0.2 yo.2. 42 = 8y. X70, yo 2 = 24 6 200y + 50g 10000 ". 100% + 50 Y - 10000 = y = 40 7 = 80 0.8 h = 20.8 5-2. 2x 2 50 0.8 -2. he 2 $ 10.8 (40)0.2 Now 0.8 400 X 2 Nmax = 10 (80) NOW we have increasing Budget 2000; ſex,y) = 100x + 50 y-12000 Fe 8xe-0-2 y0:2 +100 h,=0 -0.8 220.8 y Fy + Son= 2 → > X = 24 100x + 50 y = 12000 then 240 12000 418, 2= 96, ỹ = 250 10(96)0-8 (48)0:2 480 x 20.8 max 480 x 20.8400 x 20.8 > approximate increasing 80 x 20.8

So we have X = 80 y = 40 h = 20 0.8 5-2 Ausua N- 400 X 2 0.8 80 x 20.8 increasin = Approximate

Answer 2

1a write a function for the profit if the firm produces x units of A and y units of B

# 1b use R function nlminb to find the pair of (x,y) that maximizes the profits.

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# put your result in 'optimal1'